Properties

Label 552.2.q.d.121.1
Level $552$
Weight $2$
Character 552.121
Analytic conductor $4.408$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [552,2,Mod(25,552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(552, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("552.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.q (of order \(11\), degree \(10\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 121.1
Character \(\chi\) \(=\) 552.121
Dual form 552.2.q.d.73.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.959493 + 0.281733i) q^{3} +(-0.776140 + 0.498795i) q^{5} +(0.172807 - 1.20190i) q^{7} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\) \(q+(0.959493 + 0.281733i) q^{3} +(-0.776140 + 0.498795i) q^{5} +(0.172807 - 1.20190i) q^{7} +(0.841254 + 0.540641i) q^{9} +(1.08362 + 2.37279i) q^{11} +(0.851334 + 5.92116i) q^{13} +(-0.885227 + 0.259926i) q^{15} +(4.79395 - 5.53252i) q^{17} +(3.56144 + 4.11012i) q^{19} +(0.504420 - 1.10453i) q^{21} +(1.91294 - 4.39780i) q^{23} +(-1.72348 + 3.77389i) q^{25} +(0.654861 + 0.755750i) q^{27} +(-0.971937 + 1.12167i) q^{29} +(7.26162 - 2.13220i) q^{31} +(0.371231 + 2.58197i) q^{33} +(0.465378 + 1.01903i) q^{35} +(-6.92154 - 4.44820i) q^{37} +(-0.851334 + 5.92116i) q^{39} +(-7.88024 + 5.06433i) q^{41} +(7.55679 + 2.21887i) q^{43} -0.922599 q^{45} -10.1065 q^{47} +(5.30176 + 1.55674i) q^{49} +(6.15846 - 3.95780i) q^{51} +(-0.545566 + 3.79450i) q^{53} +(-2.02457 - 1.30111i) q^{55} +(2.25922 + 4.94701i) q^{57} +(-1.81494 - 12.6232i) q^{59} +(-5.38054 + 1.57987i) q^{61} +(0.795169 - 0.917673i) q^{63} +(-3.61420 - 4.17100i) q^{65} +(4.19459 - 9.18487i) q^{67} +(3.07446 - 3.68072i) q^{69} +(2.76110 - 6.04596i) q^{71} +(-6.43786 - 7.42969i) q^{73} +(-2.71689 + 3.13546i) q^{75} +(3.03910 - 0.892362i) q^{77} +(1.74805 + 12.1579i) q^{79} +(0.415415 + 0.909632i) q^{81} +(4.39148 + 2.82223i) q^{83} +(-0.961187 + 6.68521i) q^{85} +(-1.24858 + 0.802413i) q^{87} +(-3.57797 - 1.05059i) q^{89} +7.26374 q^{91} +7.56818 q^{93} +(-4.81428 - 1.41360i) q^{95} +(-7.58479 + 4.87445i) q^{97} +(-0.371231 + 2.58197i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 3 q^{3} + 2 q^{5} + 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 3 q^{3} + 2 q^{5} + 4 q^{7} - 3 q^{9} - 15 q^{11} - 5 q^{13} - 2 q^{15} + 9 q^{17} - 3 q^{19} + 7 q^{21} + 18 q^{23} - 19 q^{25} + 3 q^{27} - 21 q^{29} + 17 q^{31} - 7 q^{33} - 36 q^{35} + 9 q^{37} + 5 q^{39} + 18 q^{41} + 50 q^{43} + 2 q^{45} + 74 q^{47} - 17 q^{49} + 13 q^{51} + 43 q^{53} - 42 q^{55} - 8 q^{57} + 7 q^{59} - 10 q^{61} + 4 q^{63} - 4 q^{65} + 33 q^{67} + 15 q^{69} + 3 q^{71} + 30 q^{73} - 25 q^{75} - 82 q^{77} - 40 q^{79} - 3 q^{81} + 9 q^{83} - 54 q^{85} + 10 q^{87} + 25 q^{89} - 30 q^{91} + 38 q^{93} - 49 q^{95} - 69 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(185\) \(277\) \(415\)
\(\chi(n)\) \(e\left(\frac{9}{11}\right)\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.959493 + 0.281733i 0.553964 + 0.162658i
\(4\) 0 0
\(5\) −0.776140 + 0.498795i −0.347100 + 0.223068i −0.702561 0.711623i \(-0.747960\pi\)
0.355461 + 0.934691i \(0.384324\pi\)
\(6\) 0 0
\(7\) 0.172807 1.20190i 0.0653148 0.454274i −0.930751 0.365654i \(-0.880845\pi\)
0.996066 0.0886199i \(-0.0282457\pi\)
\(8\) 0 0
\(9\) 0.841254 + 0.540641i 0.280418 + 0.180214i
\(10\) 0 0
\(11\) 1.08362 + 2.37279i 0.326723 + 0.715423i 0.999706 0.0242343i \(-0.00771479\pi\)
−0.672984 + 0.739657i \(0.734988\pi\)
\(12\) 0 0
\(13\) 0.851334 + 5.92116i 0.236118 + 1.64223i 0.670791 + 0.741646i \(0.265955\pi\)
−0.434674 + 0.900588i \(0.643136\pi\)
\(14\) 0 0
\(15\) −0.885227 + 0.259926i −0.228565 + 0.0671126i
\(16\) 0 0
\(17\) 4.79395 5.53252i 1.16270 1.34183i 0.233461 0.972366i \(-0.424995\pi\)
0.929244 0.369467i \(-0.120460\pi\)
\(18\) 0 0
\(19\) 3.56144 + 4.11012i 0.817051 + 0.942927i 0.999186 0.0403469i \(-0.0128463\pi\)
−0.182135 + 0.983273i \(0.558301\pi\)
\(20\) 0 0
\(21\) 0.504420 1.10453i 0.110074 0.241027i
\(22\) 0 0
\(23\) 1.91294 4.39780i 0.398876 0.917005i
\(24\) 0 0
\(25\) −1.72348 + 3.77389i −0.344696 + 0.754778i
\(26\) 0 0
\(27\) 0.654861 + 0.755750i 0.126028 + 0.145444i
\(28\) 0 0
\(29\) −0.971937 + 1.12167i −0.180484 + 0.208290i −0.838781 0.544468i \(-0.816732\pi\)
0.658297 + 0.752758i \(0.271277\pi\)
\(30\) 0 0
\(31\) 7.26162 2.13220i 1.30423 0.382955i 0.445450 0.895307i \(-0.353044\pi\)
0.858776 + 0.512352i \(0.171226\pi\)
\(32\) 0 0
\(33\) 0.371231 + 2.58197i 0.0646230 + 0.449463i
\(34\) 0 0
\(35\) 0.465378 + 1.01903i 0.0786632 + 0.172248i
\(36\) 0 0
\(37\) −6.92154 4.44820i −1.13789 0.731280i −0.170701 0.985323i \(-0.554603\pi\)
−0.967193 + 0.254043i \(0.918239\pi\)
\(38\) 0 0
\(39\) −0.851334 + 5.92116i −0.136323 + 0.948144i
\(40\) 0 0
\(41\) −7.88024 + 5.06433i −1.23069 + 0.790915i −0.983998 0.178180i \(-0.942979\pi\)
−0.246689 + 0.969095i \(0.579343\pi\)
\(42\) 0 0
\(43\) 7.55679 + 2.21887i 1.15240 + 0.338375i 0.801475 0.598029i \(-0.204049\pi\)
0.350925 + 0.936404i \(0.385867\pi\)
\(44\) 0 0
\(45\) −0.922599 −0.137533
\(46\) 0 0
\(47\) −10.1065 −1.47419 −0.737094 0.675791i \(-0.763802\pi\)
−0.737094 + 0.675791i \(0.763802\pi\)
\(48\) 0 0
\(49\) 5.30176 + 1.55674i 0.757394 + 0.222391i
\(50\) 0 0
\(51\) 6.15846 3.95780i 0.862356 0.554203i
\(52\) 0 0
\(53\) −0.545566 + 3.79450i −0.0749393 + 0.521214i 0.917428 + 0.397901i \(0.130261\pi\)
−0.992368 + 0.123313i \(0.960648\pi\)
\(54\) 0 0
\(55\) −2.02457 1.30111i −0.272993 0.175442i
\(56\) 0 0
\(57\) 2.25922 + 4.94701i 0.299241 + 0.655247i
\(58\) 0 0
\(59\) −1.81494 12.6232i −0.236285 1.64340i −0.670009 0.742353i \(-0.733710\pi\)
0.433724 0.901046i \(-0.357199\pi\)
\(60\) 0 0
\(61\) −5.38054 + 1.57987i −0.688908 + 0.202282i −0.607406 0.794392i \(-0.707790\pi\)
−0.0815018 + 0.996673i \(0.525972\pi\)
\(62\) 0 0
\(63\) 0.795169 0.917673i 0.100182 0.115616i
\(64\) 0 0
\(65\) −3.61420 4.17100i −0.448286 0.517349i
\(66\) 0 0
\(67\) 4.19459 9.18487i 0.512451 1.12211i −0.459769 0.888039i \(-0.652068\pi\)
0.972219 0.234072i \(-0.0752051\pi\)
\(68\) 0 0
\(69\) 3.07446 3.68072i 0.370121 0.443107i
\(70\) 0 0
\(71\) 2.76110 6.04596i 0.327682 0.717524i −0.672054 0.740502i \(-0.734588\pi\)
0.999736 + 0.0229787i \(0.00731498\pi\)
\(72\) 0 0
\(73\) −6.43786 7.42969i −0.753495 0.869579i 0.241408 0.970424i \(-0.422391\pi\)
−0.994902 + 0.100845i \(0.967845\pi\)
\(74\) 0 0
\(75\) −2.71689 + 3.13546i −0.313720 + 0.362052i
\(76\) 0 0
\(77\) 3.03910 0.892362i 0.346338 0.101694i
\(78\) 0 0
\(79\) 1.74805 + 12.1579i 0.196671 + 1.36787i 0.813860 + 0.581060i \(0.197362\pi\)
−0.617190 + 0.786814i \(0.711729\pi\)
\(80\) 0 0
\(81\) 0.415415 + 0.909632i 0.0461572 + 0.101070i
\(82\) 0 0
\(83\) 4.39148 + 2.82223i 0.482027 + 0.309780i 0.758992 0.651100i \(-0.225692\pi\)
−0.276965 + 0.960880i \(0.589329\pi\)
\(84\) 0 0
\(85\) −0.961187 + 6.68521i −0.104255 + 0.725112i
\(86\) 0 0
\(87\) −1.24858 + 0.802413i −0.133862 + 0.0860277i
\(88\) 0 0
\(89\) −3.57797 1.05059i −0.379264 0.111362i 0.0865433 0.996248i \(-0.472418\pi\)
−0.465807 + 0.884886i \(0.654236\pi\)
\(90\) 0 0
\(91\) 7.26374 0.761447
\(92\) 0 0
\(93\) 7.56818 0.784784
\(94\) 0 0
\(95\) −4.81428 1.41360i −0.493935 0.145032i
\(96\) 0 0
\(97\) −7.58479 + 4.87445i −0.770118 + 0.494925i −0.865741 0.500493i \(-0.833152\pi\)
0.0956225 + 0.995418i \(0.469516\pi\)
\(98\) 0 0
\(99\) −0.371231 + 2.58197i −0.0373101 + 0.259497i
\(100\) 0 0
\(101\) −10.7948 6.93742i −1.07413 0.690299i −0.120933 0.992661i \(-0.538588\pi\)
−0.953193 + 0.302362i \(0.902225\pi\)
\(102\) 0 0
\(103\) −2.70610 5.92552i −0.266639 0.583859i 0.728195 0.685370i \(-0.240360\pi\)
−0.994834 + 0.101511i \(0.967632\pi\)
\(104\) 0 0
\(105\) 0.159431 + 1.10887i 0.0155589 + 0.108214i
\(106\) 0 0
\(107\) 0.439592 0.129076i 0.0424969 0.0124782i −0.260415 0.965497i \(-0.583859\pi\)
0.302912 + 0.953019i \(0.402041\pi\)
\(108\) 0 0
\(109\) −6.01515 + 6.94185i −0.576147 + 0.664909i −0.966772 0.255641i \(-0.917713\pi\)
0.390625 + 0.920550i \(0.372259\pi\)
\(110\) 0 0
\(111\) −5.38796 6.21804i −0.511403 0.590190i
\(112\) 0 0
\(113\) 0.354339 0.775894i 0.0333334 0.0729900i −0.892233 0.451576i \(-0.850862\pi\)
0.925566 + 0.378586i \(0.123589\pi\)
\(114\) 0 0
\(115\) 0.708891 + 4.36747i 0.0661044 + 0.407269i
\(116\) 0 0
\(117\) −2.48503 + 5.44146i −0.229741 + 0.503063i
\(118\) 0 0
\(119\) −5.82109 6.71789i −0.533618 0.615828i
\(120\) 0 0
\(121\) 2.74756 3.17085i 0.249778 0.288259i
\(122\) 0 0
\(123\) −8.98782 + 2.63906i −0.810405 + 0.237956i
\(124\) 0 0
\(125\) −1.20123 8.35477i −0.107442 0.747273i
\(126\) 0 0
\(127\) −5.55691 12.1679i −0.493096 1.07973i −0.978652 0.205525i \(-0.934110\pi\)
0.485556 0.874206i \(-0.338617\pi\)
\(128\) 0 0
\(129\) 6.62556 + 4.25799i 0.583348 + 0.374895i
\(130\) 0 0
\(131\) 0.635443 4.41961i 0.0555189 0.386143i −0.943049 0.332653i \(-0.892056\pi\)
0.998568 0.0534901i \(-0.0170346\pi\)
\(132\) 0 0
\(133\) 5.55538 3.57023i 0.481713 0.309578i
\(134\) 0 0
\(135\) −0.885227 0.259926i −0.0761882 0.0223709i
\(136\) 0 0
\(137\) 19.5982 1.67439 0.837193 0.546907i \(-0.184195\pi\)
0.837193 + 0.546907i \(0.184195\pi\)
\(138\) 0 0
\(139\) 0.590626 0.0500962 0.0250481 0.999686i \(-0.492026\pi\)
0.0250481 + 0.999686i \(0.492026\pi\)
\(140\) 0 0
\(141\) −9.69713 2.84734i −0.816646 0.239789i
\(142\) 0 0
\(143\) −13.1271 + 8.43631i −1.09775 + 0.705479i
\(144\) 0 0
\(145\) 0.194873 1.35537i 0.0161833 0.112558i
\(146\) 0 0
\(147\) 4.64842 + 2.98736i 0.383395 + 0.246393i
\(148\) 0 0
\(149\) −7.36229 16.1212i −0.603142 1.32070i −0.927167 0.374648i \(-0.877763\pi\)
0.324025 0.946049i \(-0.394964\pi\)
\(150\) 0 0
\(151\) −2.06189 14.3408i −0.167794 1.16703i −0.883431 0.468562i \(-0.844772\pi\)
0.715637 0.698473i \(-0.246137\pi\)
\(152\) 0 0
\(153\) 7.02404 2.06244i 0.567860 0.166739i
\(154\) 0 0
\(155\) −4.57250 + 5.27694i −0.367272 + 0.423854i
\(156\) 0 0
\(157\) −0.237882 0.274530i −0.0189850 0.0219099i 0.746178 0.665747i \(-0.231887\pi\)
−0.765163 + 0.643837i \(0.777341\pi\)
\(158\) 0 0
\(159\) −1.59250 + 3.48709i −0.126293 + 0.276544i
\(160\) 0 0
\(161\) −4.95513 3.05913i −0.390519 0.241093i
\(162\) 0 0
\(163\) −4.77773 + 10.4618i −0.374221 + 0.819429i 0.625025 + 0.780604i \(0.285089\pi\)
−0.999246 + 0.0388246i \(0.987639\pi\)
\(164\) 0 0
\(165\) −1.57600 1.81880i −0.122691 0.141593i
\(166\) 0 0
\(167\) −3.08530 + 3.56063i −0.238748 + 0.275530i −0.862461 0.506124i \(-0.831078\pi\)
0.623713 + 0.781653i \(0.285623\pi\)
\(168\) 0 0
\(169\) −21.8619 + 6.41925i −1.68169 + 0.493788i
\(170\) 0 0
\(171\) 0.773975 + 5.38311i 0.0591873 + 0.411657i
\(172\) 0 0
\(173\) −3.22786 7.06803i −0.245410 0.537372i 0.746339 0.665565i \(-0.231810\pi\)
−0.991749 + 0.128193i \(0.959082\pi\)
\(174\) 0 0
\(175\) 4.23800 + 2.72360i 0.320363 + 0.205885i
\(176\) 0 0
\(177\) 1.81494 12.6232i 0.136419 0.948816i
\(178\) 0 0
\(179\) −12.4407 + 7.99518i −0.929865 + 0.597588i −0.915504 0.402309i \(-0.868208\pi\)
−0.0143609 + 0.999897i \(0.504571\pi\)
\(180\) 0 0
\(181\) 18.1390 + 5.32611i 1.34826 + 0.395886i 0.874611 0.484825i \(-0.161117\pi\)
0.473654 + 0.880711i \(0.342935\pi\)
\(182\) 0 0
\(183\) −5.60769 −0.414533
\(184\) 0 0
\(185\) 7.59082 0.558088
\(186\) 0 0
\(187\) 18.3223 + 5.37992i 1.33986 + 0.393419i
\(188\) 0 0
\(189\) 1.02150 0.656476i 0.0743030 0.0477516i
\(190\) 0 0
\(191\) 0.0524728 0.364957i 0.00379680 0.0264073i −0.987835 0.155506i \(-0.950299\pi\)
0.991632 + 0.129099i \(0.0412083\pi\)
\(192\) 0 0
\(193\) −16.7208 10.7458i −1.20359 0.773499i −0.224014 0.974586i \(-0.571916\pi\)
−0.979573 + 0.201087i \(0.935553\pi\)
\(194\) 0 0
\(195\) −2.29269 5.02029i −0.164183 0.359510i
\(196\) 0 0
\(197\) −0.527469 3.66863i −0.0375806 0.261379i 0.962366 0.271758i \(-0.0876052\pi\)
−0.999946 + 0.0103794i \(0.996696\pi\)
\(198\) 0 0
\(199\) −6.97900 + 2.04922i −0.494728 + 0.145265i −0.519573 0.854426i \(-0.673909\pi\)
0.0248450 + 0.999691i \(0.492091\pi\)
\(200\) 0 0
\(201\) 6.61236 7.63107i 0.466400 0.538254i
\(202\) 0 0
\(203\) 1.18018 + 1.36200i 0.0828324 + 0.0955937i
\(204\) 0 0
\(205\) 3.59011 7.86125i 0.250744 0.549053i
\(206\) 0 0
\(207\) 3.98690 2.66545i 0.277109 0.185262i
\(208\) 0 0
\(209\) −5.89322 + 12.9043i −0.407642 + 0.892612i
\(210\) 0 0
\(211\) −0.678026 0.782483i −0.0466772 0.0538684i 0.731931 0.681379i \(-0.238619\pi\)
−0.778608 + 0.627511i \(0.784074\pi\)
\(212\) 0 0
\(213\) 4.35260 5.02317i 0.298235 0.344182i
\(214\) 0 0
\(215\) −6.97188 + 2.04713i −0.475479 + 0.139613i
\(216\) 0 0
\(217\) −1.30783 9.09618i −0.0887815 0.617489i
\(218\) 0 0
\(219\) −4.08390 8.94249i −0.275964 0.604277i
\(220\) 0 0
\(221\) 36.8402 + 23.6757i 2.47814 + 1.59260i
\(222\) 0 0
\(223\) 0.0839960 0.584205i 0.00562479 0.0391213i −0.986816 0.161848i \(-0.948255\pi\)
0.992441 + 0.122726i \(0.0391638\pi\)
\(224\) 0 0
\(225\) −3.49020 + 2.24302i −0.232680 + 0.149534i
\(226\) 0 0
\(227\) 17.7311 + 5.20632i 1.17685 + 0.345556i 0.810960 0.585101i \(-0.198945\pi\)
0.365894 + 0.930657i \(0.380763\pi\)
\(228\) 0 0
\(229\) 0.328364 0.0216989 0.0108494 0.999941i \(-0.496546\pi\)
0.0108494 + 0.999941i \(0.496546\pi\)
\(230\) 0 0
\(231\) 3.16741 0.208400
\(232\) 0 0
\(233\) 17.8023 + 5.22723i 1.16627 + 0.342448i 0.806866 0.590735i \(-0.201162\pi\)
0.359403 + 0.933182i \(0.382980\pi\)
\(234\) 0 0
\(235\) 7.84407 5.04108i 0.511691 0.328844i
\(236\) 0 0
\(237\) −1.74805 + 12.1579i −0.113548 + 0.789743i
\(238\) 0 0
\(239\) −4.08514 2.62536i −0.264246 0.169820i 0.401809 0.915724i \(-0.368382\pi\)
−0.666054 + 0.745903i \(0.732018\pi\)
\(240\) 0 0
\(241\) 11.1472 + 24.4089i 0.718053 + 1.57232i 0.816613 + 0.577186i \(0.195849\pi\)
−0.0985599 + 0.995131i \(0.531424\pi\)
\(242\) 0 0
\(243\) 0.142315 + 0.989821i 0.00912950 + 0.0634971i
\(244\) 0 0
\(245\) −4.89140 + 1.43624i −0.312500 + 0.0917582i
\(246\) 0 0
\(247\) −21.3047 + 24.5869i −1.35559 + 1.56443i
\(248\) 0 0
\(249\) 3.41848 + 3.94513i 0.216637 + 0.250013i
\(250\) 0 0
\(251\) −0.513571 + 1.12456i −0.0324163 + 0.0709818i −0.925148 0.379608i \(-0.876059\pi\)
0.892731 + 0.450590i \(0.148786\pi\)
\(252\) 0 0
\(253\) 12.5080 0.226526i 0.786368 0.0142416i
\(254\) 0 0
\(255\) −2.80569 + 6.14361i −0.175699 + 0.384728i
\(256\) 0 0
\(257\) −0.0850846 0.0981929i −0.00530743 0.00612510i 0.753090 0.657918i \(-0.228563\pi\)
−0.758397 + 0.651793i \(0.774017\pi\)
\(258\) 0 0
\(259\) −6.54237 + 7.55029i −0.406523 + 0.469152i
\(260\) 0 0
\(261\) −1.42407 + 0.418144i −0.0881476 + 0.0258825i
\(262\) 0 0
\(263\) 0.0268550 + 0.186781i 0.00165595 + 0.0115174i 0.990633 0.136554i \(-0.0436026\pi\)
−0.988977 + 0.148071i \(0.952694\pi\)
\(264\) 0 0
\(265\) −1.46924 3.21718i −0.0902546 0.197630i
\(266\) 0 0
\(267\) −3.13705 2.01606i −0.191985 0.123381i
\(268\) 0 0
\(269\) −3.05392 + 21.2405i −0.186201 + 1.29506i 0.655534 + 0.755165i \(0.272443\pi\)
−0.841735 + 0.539891i \(0.818466\pi\)
\(270\) 0 0
\(271\) 2.96855 1.90777i 0.180327 0.115889i −0.447363 0.894352i \(-0.647637\pi\)
0.627690 + 0.778463i \(0.284001\pi\)
\(272\) 0 0
\(273\) 6.96950 + 2.04643i 0.421814 + 0.123856i
\(274\) 0 0
\(275\) −10.8222 −0.652606
\(276\) 0 0
\(277\) 18.6506 1.12061 0.560303 0.828288i \(-0.310685\pi\)
0.560303 + 0.828288i \(0.310685\pi\)
\(278\) 0 0
\(279\) 7.26162 + 2.13220i 0.434742 + 0.127652i
\(280\) 0 0
\(281\) 22.8746 14.7006i 1.36458 0.876964i 0.366022 0.930606i \(-0.380719\pi\)
0.998560 + 0.0536420i \(0.0170830\pi\)
\(282\) 0 0
\(283\) 0.528675 3.67701i 0.0314264 0.218576i −0.968056 0.250733i \(-0.919329\pi\)
0.999483 + 0.0321569i \(0.0102376\pi\)
\(284\) 0 0
\(285\) −4.22101 2.71268i −0.250031 0.160685i
\(286\) 0 0
\(287\) 4.72504 + 10.3464i 0.278910 + 0.610728i
\(288\) 0 0
\(289\) −5.20741 36.2183i −0.306318 2.13049i
\(290\) 0 0
\(291\) −8.65084 + 2.54012i −0.507121 + 0.148904i
\(292\) 0 0
\(293\) 11.0833 12.7908i 0.647494 0.747248i −0.333187 0.942861i \(-0.608124\pi\)
0.980681 + 0.195613i \(0.0626696\pi\)
\(294\) 0 0
\(295\) 7.70502 + 8.89207i 0.448604 + 0.517716i
\(296\) 0 0
\(297\) −1.08362 + 2.37279i −0.0628778 + 0.137683i
\(298\) 0 0
\(299\) 27.6686 + 7.58283i 1.60012 + 0.438526i
\(300\) 0 0
\(301\) 3.97272 8.69904i 0.228984 0.501404i
\(302\) 0 0
\(303\) −8.40307 9.69766i −0.482744 0.557116i
\(304\) 0 0
\(305\) 3.38802 3.90998i 0.193998 0.223885i
\(306\) 0 0
\(307\) 27.5809 8.09848i 1.57412 0.462204i 0.625925 0.779883i \(-0.284721\pi\)
0.948198 + 0.317679i \(0.102903\pi\)
\(308\) 0 0
\(309\) −0.927067 6.44789i −0.0527390 0.366808i
\(310\) 0 0
\(311\) 10.7308 + 23.4972i 0.608488 + 1.33240i 0.923603 + 0.383350i \(0.125230\pi\)
−0.315115 + 0.949053i \(0.602043\pi\)
\(312\) 0 0
\(313\) 9.80550 + 6.30161i 0.554240 + 0.356188i 0.787585 0.616206i \(-0.211331\pi\)
−0.233346 + 0.972394i \(0.574967\pi\)
\(314\) 0 0
\(315\) −0.159431 + 1.10887i −0.00898293 + 0.0624777i
\(316\) 0 0
\(317\) −13.0585 + 8.39220i −0.733440 + 0.471353i −0.853288 0.521439i \(-0.825395\pi\)
0.119849 + 0.992792i \(0.461759\pi\)
\(318\) 0 0
\(319\) −3.71471 1.09074i −0.207984 0.0610695i
\(320\) 0 0
\(321\) 0.458150 0.0255714
\(322\) 0 0
\(323\) 39.8127 2.21524
\(324\) 0 0
\(325\) −23.8131 6.99215i −1.32091 0.387855i
\(326\) 0 0
\(327\) −7.72724 + 4.96599i −0.427317 + 0.274620i
\(328\) 0 0
\(329\) −1.74647 + 12.1470i −0.0962862 + 0.669685i
\(330\) 0 0
\(331\) −26.4789 17.0169i −1.45541 0.935335i −0.998960 0.0455860i \(-0.985484\pi\)
−0.456450 0.889749i \(-0.650879\pi\)
\(332\) 0 0
\(333\) −3.41789 7.48413i −0.187299 0.410128i
\(334\) 0 0
\(335\) 1.32578 + 9.22098i 0.0724349 + 0.503796i
\(336\) 0 0
\(337\) −18.1231 + 5.32144i −0.987231 + 0.289877i −0.735206 0.677844i \(-0.762915\pi\)
−0.252025 + 0.967721i \(0.581096\pi\)
\(338\) 0 0
\(339\) 0.558581 0.644636i 0.0303379 0.0350118i
\(340\) 0 0
\(341\) 12.9281 + 14.9198i 0.700095 + 0.807953i
\(342\) 0 0
\(343\) 6.31816 13.8348i 0.341148 0.747011i
\(344\) 0 0
\(345\) −0.550283 + 4.39028i −0.0296263 + 0.236365i
\(346\) 0 0
\(347\) −8.48901 + 18.5883i −0.455714 + 0.997874i 0.532730 + 0.846285i \(0.321166\pi\)
−0.988444 + 0.151589i \(0.951561\pi\)
\(348\) 0 0
\(349\) 21.6680 + 25.0062i 1.15986 + 1.33855i 0.930967 + 0.365102i \(0.118966\pi\)
0.228896 + 0.973451i \(0.426489\pi\)
\(350\) 0 0
\(351\) −3.91741 + 4.52093i −0.209096 + 0.241309i
\(352\) 0 0
\(353\) −4.81493 + 1.41379i −0.256273 + 0.0752485i −0.407345 0.913274i \(-0.633545\pi\)
0.151073 + 0.988523i \(0.451727\pi\)
\(354\) 0 0
\(355\) 0.872695 + 6.06973i 0.0463179 + 0.322148i
\(356\) 0 0
\(357\) −3.69264 8.08576i −0.195435 0.427944i
\(358\) 0 0
\(359\) −25.5448 16.4167i −1.34820 0.866438i −0.350662 0.936502i \(-0.614043\pi\)
−0.997543 + 0.0700638i \(0.977680\pi\)
\(360\) 0 0
\(361\) −1.50526 + 10.4693i −0.0792241 + 0.551016i
\(362\) 0 0
\(363\) 3.52960 2.26833i 0.185256 0.119057i
\(364\) 0 0
\(365\) 8.70257 + 2.55530i 0.455513 + 0.133751i
\(366\) 0 0
\(367\) 9.52268 0.497080 0.248540 0.968622i \(-0.420049\pi\)
0.248540 + 0.968622i \(0.420049\pi\)
\(368\) 0 0
\(369\) −9.36726 −0.487640
\(370\) 0 0
\(371\) 4.46632 + 1.31143i 0.231880 + 0.0680860i
\(372\) 0 0
\(373\) 4.07637 2.61972i 0.211067 0.135644i −0.430837 0.902430i \(-0.641781\pi\)
0.641903 + 0.766786i \(0.278145\pi\)
\(374\) 0 0
\(375\) 1.20123 8.35477i 0.0620315 0.431438i
\(376\) 0 0
\(377\) −7.46906 4.80007i −0.384676 0.247216i
\(378\) 0 0
\(379\) −11.9353 26.1348i −0.613077 1.34245i −0.920449 0.390863i \(-0.872177\pi\)
0.307372 0.951590i \(-0.400551\pi\)
\(380\) 0 0
\(381\) −1.90371 13.2406i −0.0975302 0.678337i
\(382\) 0 0
\(383\) −23.4241 + 6.87794i −1.19692 + 0.351446i −0.818673 0.574260i \(-0.805290\pi\)
−0.378243 + 0.925706i \(0.623472\pi\)
\(384\) 0 0
\(385\) −1.91366 + 2.20849i −0.0975294 + 0.112555i
\(386\) 0 0
\(387\) 5.15756 + 5.95214i 0.262174 + 0.302564i
\(388\) 0 0
\(389\) 0.102888 0.225294i 0.00521664 0.0114229i −0.907006 0.421118i \(-0.861638\pi\)
0.912222 + 0.409695i \(0.134365\pi\)
\(390\) 0 0
\(391\) −15.1604 31.6662i −0.766693 1.60143i
\(392\) 0 0
\(393\) 1.85485 4.06155i 0.0935648 0.204878i
\(394\) 0 0
\(395\) −7.42104 8.56434i −0.373393 0.430919i
\(396\) 0 0
\(397\) 2.55739 2.95139i 0.128352 0.148126i −0.687936 0.725772i \(-0.741483\pi\)
0.816288 + 0.577646i \(0.196028\pi\)
\(398\) 0 0
\(399\) 6.33620 1.86048i 0.317207 0.0931403i
\(400\) 0 0
\(401\) −0.491431 3.41798i −0.0245409 0.170686i 0.973865 0.227128i \(-0.0729335\pi\)
−0.998406 + 0.0564420i \(0.982024\pi\)
\(402\) 0 0
\(403\) 18.8072 + 41.1820i 0.936852 + 2.05142i
\(404\) 0 0
\(405\) −0.776140 0.498795i −0.0385667 0.0247853i
\(406\) 0 0
\(407\) 3.05435 21.2435i 0.151399 1.05300i
\(408\) 0 0
\(409\) −18.6949 + 12.0145i −0.924403 + 0.594078i −0.913932 0.405867i \(-0.866969\pi\)
−0.0104711 + 0.999945i \(0.503333\pi\)
\(410\) 0 0
\(411\) 18.8043 + 5.52145i 0.927549 + 0.272353i
\(412\) 0 0
\(413\) −15.4854 −0.761986
\(414\) 0 0
\(415\) −4.81611 −0.236414
\(416\) 0 0
\(417\) 0.566701 + 0.166398i 0.0277515 + 0.00814857i
\(418\) 0 0
\(419\) −1.04864 + 0.673918i −0.0512292 + 0.0329230i −0.566005 0.824402i \(-0.691512\pi\)
0.514776 + 0.857325i \(0.327875\pi\)
\(420\) 0 0
\(421\) −3.44772 + 23.9794i −0.168031 + 1.16868i 0.714915 + 0.699211i \(0.246465\pi\)
−0.882947 + 0.469473i \(0.844444\pi\)
\(422\) 0 0
\(423\) −8.50214 5.46400i −0.413388 0.265669i
\(424\) 0 0
\(425\) 12.6168 + 27.6270i 0.612007 + 1.34011i
\(426\) 0 0
\(427\) 0.969047 + 6.73987i 0.0468954 + 0.326165i
\(428\) 0 0
\(429\) −14.9722 + 4.39623i −0.722864 + 0.212252i
\(430\) 0 0
\(431\) 17.8204 20.5658i 0.858378 0.990622i −0.141621 0.989921i \(-0.545232\pi\)
1.00000 0.000700669i \(-0.000223030\pi\)
\(432\) 0 0
\(433\) −23.8363 27.5086i −1.14550 1.32198i −0.939153 0.343500i \(-0.888387\pi\)
−0.206349 0.978479i \(-0.566158\pi\)
\(434\) 0 0
\(435\) 0.568832 1.24557i 0.0272734 0.0597205i
\(436\) 0 0
\(437\) 24.8883 7.80009i 1.19057 0.373129i
\(438\) 0 0
\(439\) −2.41170 + 5.28088i −0.115104 + 0.252043i −0.958412 0.285388i \(-0.907878\pi\)
0.843308 + 0.537430i \(0.180605\pi\)
\(440\) 0 0
\(441\) 3.61849 + 4.17596i 0.172309 + 0.198855i
\(442\) 0 0
\(443\) −14.3787 + 16.5939i −0.683153 + 0.788401i −0.986374 0.164519i \(-0.947393\pi\)
0.303220 + 0.952920i \(0.401938\pi\)
\(444\) 0 0
\(445\) 3.30103 0.969270i 0.156484 0.0459478i
\(446\) 0 0
\(447\) −2.52221 17.5423i −0.119296 0.829724i
\(448\) 0 0
\(449\) 0.558752 + 1.22350i 0.0263691 + 0.0577403i 0.922358 0.386336i \(-0.126259\pi\)
−0.895989 + 0.444076i \(0.853532\pi\)
\(450\) 0 0
\(451\) −20.5557 13.2104i −0.967932 0.622052i
\(452\) 0 0
\(453\) 2.06189 14.3408i 0.0968761 0.673788i
\(454\) 0 0
\(455\) −5.63767 + 3.62311i −0.264298 + 0.169854i
\(456\) 0 0
\(457\) −15.4742 4.54364i −0.723854 0.212543i −0.101006 0.994886i \(-0.532206\pi\)
−0.622848 + 0.782343i \(0.714024\pi\)
\(458\) 0 0
\(459\) 7.32057 0.341695
\(460\) 0 0
\(461\) −22.4689 −1.04648 −0.523240 0.852185i \(-0.675277\pi\)
−0.523240 + 0.852185i \(0.675277\pi\)
\(462\) 0 0
\(463\) 24.4326 + 7.17405i 1.13548 + 0.333406i 0.794859 0.606795i \(-0.207545\pi\)
0.340620 + 0.940201i \(0.389363\pi\)
\(464\) 0 0
\(465\) −5.87397 + 3.77497i −0.272399 + 0.175060i
\(466\) 0 0
\(467\) 1.78966 12.4474i 0.0828155 0.575995i −0.905590 0.424155i \(-0.860571\pi\)
0.988405 0.151840i \(-0.0485198\pi\)
\(468\) 0 0
\(469\) −10.3144 6.62867i −0.476275 0.306084i
\(470\) 0 0
\(471\) −0.150902 0.330429i −0.00695319 0.0152254i
\(472\) 0 0
\(473\) 2.92374 + 20.3351i 0.134434 + 0.935008i
\(474\) 0 0
\(475\) −21.6492 + 6.35678i −0.993334 + 0.291669i
\(476\) 0 0
\(477\) −2.51042 + 2.89718i −0.114944 + 0.132653i
\(478\) 0 0
\(479\) 16.8571 + 19.4541i 0.770221 + 0.888883i 0.996363 0.0852117i \(-0.0271567\pi\)
−0.226142 + 0.974094i \(0.572611\pi\)
\(480\) 0 0
\(481\) 20.4460 44.7704i 0.932256 2.04136i
\(482\) 0 0
\(483\) −3.89256 4.33123i −0.177118 0.197078i
\(484\) 0 0
\(485\) 3.45550 7.56650i 0.156906 0.343577i
\(486\) 0 0
\(487\) −11.9101 13.7450i −0.539697 0.622844i 0.418754 0.908100i \(-0.362467\pi\)
−0.958451 + 0.285256i \(0.907921\pi\)
\(488\) 0 0
\(489\) −7.53162 + 8.69195i −0.340592 + 0.393064i
\(490\) 0 0
\(491\) −6.90525 + 2.02757i −0.311630 + 0.0915027i −0.433808 0.901005i \(-0.642830\pi\)
0.122178 + 0.992508i \(0.461012\pi\)
\(492\) 0 0
\(493\) 1.54627 + 10.7545i 0.0696403 + 0.484359i
\(494\) 0 0
\(495\) −0.999744 2.18913i −0.0449351 0.0983942i
\(496\) 0 0
\(497\) −6.78948 4.36334i −0.304550 0.195722i
\(498\) 0 0
\(499\) 1.61778 11.2519i 0.0724217 0.503704i −0.921034 0.389483i \(-0.872654\pi\)
0.993455 0.114221i \(-0.0364372\pi\)
\(500\) 0 0
\(501\) −3.96347 + 2.54717i −0.177075 + 0.113799i
\(502\) 0 0
\(503\) 38.2992 + 11.2456i 1.70767 + 0.501419i 0.982360 0.186999i \(-0.0598761\pi\)
0.725315 + 0.688417i \(0.241694\pi\)
\(504\) 0 0
\(505\) 11.8386 0.526813
\(506\) 0 0
\(507\) −22.7849 −1.01191
\(508\) 0 0
\(509\) −13.9259 4.08902i −0.617255 0.181243i −0.0418674 0.999123i \(-0.513331\pi\)
−0.575388 + 0.817881i \(0.695149\pi\)
\(510\) 0 0
\(511\) −10.0422 + 6.45374i −0.444242 + 0.285497i
\(512\) 0 0
\(513\) −0.773975 + 5.38311i −0.0341718 + 0.237670i
\(514\) 0 0
\(515\) 5.05593 + 3.24925i 0.222791 + 0.143179i
\(516\) 0 0
\(517\) −10.9516 23.9806i −0.481651 1.05467i
\(518\) 0 0
\(519\) −1.10582 7.69112i −0.0485399 0.337603i
\(520\) 0 0
\(521\) −22.7207 + 6.67139i −0.995411 + 0.292279i −0.738571 0.674175i \(-0.764499\pi\)
−0.256839 + 0.966454i \(0.582681\pi\)
\(522\) 0 0
\(523\) 16.0912 18.5703i 0.703620 0.812021i −0.285617 0.958344i \(-0.592198\pi\)
0.989237 + 0.146323i \(0.0467438\pi\)
\(524\) 0 0
\(525\) 3.29900 + 3.80725i 0.143980 + 0.166162i
\(526\) 0 0
\(527\) 23.0154 50.3967i 1.00257 2.19532i
\(528\) 0 0
\(529\) −15.6813 16.8255i −0.681796 0.731542i
\(530\) 0 0
\(531\) 5.29778 11.6005i 0.229904 0.503420i
\(532\) 0 0
\(533\) −36.6954 42.3487i −1.58945 1.83433i
\(534\) 0 0
\(535\) −0.276802 + 0.319447i −0.0119672 + 0.0138109i
\(536\) 0 0
\(537\) −14.1893 + 4.16636i −0.612314 + 0.179792i
\(538\) 0 0
\(539\) 2.05127 + 14.2669i 0.0883543 + 0.614517i
\(540\) 0 0
\(541\) 9.70107 + 21.2424i 0.417082 + 0.913282i 0.995249 + 0.0973633i \(0.0310409\pi\)
−0.578167 + 0.815918i \(0.696232\pi\)
\(542\) 0 0
\(543\) 15.9038 + 10.2207i 0.682495 + 0.438613i
\(544\) 0 0
\(545\) 1.20604 8.38817i 0.0516609 0.359310i
\(546\) 0 0
\(547\) −20.6404 + 13.2648i −0.882519 + 0.567160i −0.901558 0.432658i \(-0.857576\pi\)
0.0190391 + 0.999819i \(0.493939\pi\)
\(548\) 0 0
\(549\) −5.38054 1.57987i −0.229636 0.0674272i
\(550\) 0 0
\(551\) −8.07172 −0.343867
\(552\) 0 0
\(553\) 14.9147 0.634236
\(554\) 0 0
\(555\) 7.28334 + 2.13858i 0.309160 + 0.0907777i
\(556\) 0 0
\(557\) −1.35322 + 0.869662i −0.0573378 + 0.0368488i −0.568996 0.822341i \(-0.692668\pi\)
0.511658 + 0.859189i \(0.329032\pi\)
\(558\) 0 0
\(559\) −6.70495 + 46.6339i −0.283589 + 1.97241i
\(560\) 0 0
\(561\) 16.0644 + 10.3240i 0.678241 + 0.435879i
\(562\) 0 0
\(563\) −1.66197 3.63920i −0.0700435 0.153374i 0.871372 0.490623i \(-0.163231\pi\)
−0.941415 + 0.337249i \(0.890504\pi\)
\(564\) 0 0
\(565\) 0.111995 + 0.778945i 0.00471168 + 0.0327705i
\(566\) 0 0
\(567\) 1.16507 0.342095i 0.0489283 0.0143667i
\(568\) 0 0
\(569\) −20.4197 + 23.5656i −0.856039 + 0.987921i −0.999999 0.00166215i \(-0.999471\pi\)
0.143960 + 0.989584i \(0.454016\pi\)
\(570\) 0 0
\(571\) −2.05210 2.36825i −0.0858777 0.0991082i 0.711184 0.703006i \(-0.248159\pi\)
−0.797062 + 0.603897i \(0.793614\pi\)
\(572\) 0 0
\(573\) 0.153167 0.335390i 0.00639866 0.0140111i
\(574\) 0 0
\(575\) 13.2999 + 14.7987i 0.554645 + 0.617150i
\(576\) 0 0
\(577\) −0.484755 + 1.06147i −0.0201806 + 0.0441894i −0.919454 0.393198i \(-0.871369\pi\)
0.899273 + 0.437387i \(0.144096\pi\)
\(578\) 0 0
\(579\) −13.0160 15.0213i −0.540928 0.624264i
\(580\) 0 0
\(581\) 4.15091 4.79040i 0.172209 0.198739i
\(582\) 0 0
\(583\) −9.59473 + 2.81727i −0.397373 + 0.116679i
\(584\) 0 0
\(585\) −0.785440 5.46285i −0.0324739 0.225861i
\(586\) 0 0
\(587\) −5.12710 11.2268i −0.211618 0.463379i 0.773822 0.633403i \(-0.218343\pi\)
−0.985440 + 0.170024i \(0.945615\pi\)
\(588\) 0 0
\(589\) 34.6254 + 22.2524i 1.42672 + 0.916895i
\(590\) 0 0
\(591\) 0.527469 3.66863i 0.0216972 0.150907i
\(592\) 0 0
\(593\) 6.59087 4.23569i 0.270655 0.173939i −0.398273 0.917267i \(-0.630390\pi\)
0.668927 + 0.743328i \(0.266754\pi\)
\(594\) 0 0
\(595\) 7.86883 + 2.31050i 0.322590 + 0.0947211i
\(596\) 0 0
\(597\) −7.27363 −0.297690
\(598\) 0 0
\(599\) −24.6991 −1.00918 −0.504590 0.863359i \(-0.668356\pi\)
−0.504590 + 0.863359i \(0.668356\pi\)
\(600\) 0 0
\(601\) −13.1925 3.87365i −0.538131 0.158010i 0.00136264 0.999999i \(-0.499566\pi\)
−0.539494 + 0.841989i \(0.681384\pi\)
\(602\) 0 0
\(603\) 8.49443 5.45904i 0.345920 0.222309i
\(604\) 0 0
\(605\) −0.550886 + 3.83149i −0.0223967 + 0.155772i
\(606\) 0 0
\(607\) −8.35547 5.36973i −0.339138 0.217951i 0.359975 0.932962i \(-0.382785\pi\)
−0.699113 + 0.715011i \(0.746422\pi\)
\(608\) 0 0
\(609\) 0.748655 + 1.63933i 0.0303370 + 0.0664288i
\(610\) 0 0
\(611\) −8.60402 59.8423i −0.348082 2.42096i
\(612\) 0 0
\(613\) 16.1328 4.73701i 0.651596 0.191326i 0.0608039 0.998150i \(-0.480634\pi\)
0.590792 + 0.806824i \(0.298815\pi\)
\(614\) 0 0
\(615\) 5.65946 6.53136i 0.228211 0.263370i
\(616\) 0 0
\(617\) 10.7749 + 12.4350i 0.433783 + 0.500612i 0.929986 0.367594i \(-0.119818\pi\)
−0.496203 + 0.868206i \(0.665273\pi\)
\(618\) 0 0
\(619\) 3.43906 7.53049i 0.138227 0.302676i −0.827841 0.560963i \(-0.810431\pi\)
0.966068 + 0.258287i \(0.0831581\pi\)
\(620\) 0 0
\(621\) 4.57635 1.43424i 0.183642 0.0575542i
\(622\) 0 0
\(623\) −1.88099 + 4.11880i −0.0753604 + 0.165016i
\(624\) 0 0
\(625\) −8.48483 9.79201i −0.339393 0.391681i
\(626\) 0 0
\(627\) −9.29008 + 10.7213i −0.371010 + 0.428168i
\(628\) 0 0
\(629\) −57.7913 + 16.9691i −2.30429 + 0.676600i
\(630\) 0 0
\(631\) −4.21752 29.3335i −0.167897 1.16775i −0.883222 0.468955i \(-0.844631\pi\)
0.715325 0.698792i \(-0.246279\pi\)
\(632\) 0 0
\(633\) −0.430110 0.941809i −0.0170953 0.0374336i
\(634\) 0 0
\(635\) 10.3822 + 6.67226i 0.412007 + 0.264781i
\(636\) 0 0
\(637\) −4.70412 + 32.7179i −0.186384 + 1.29633i
\(638\) 0 0
\(639\) 5.59148 3.59342i 0.221195 0.142154i
\(640\) 0 0
\(641\) 2.23010 + 0.654815i 0.0880835 + 0.0258637i 0.325477 0.945550i \(-0.394475\pi\)
−0.237394 + 0.971413i \(0.576293\pi\)
\(642\) 0 0
\(643\) 5.66168 0.223275 0.111637 0.993749i \(-0.464390\pi\)
0.111637 + 0.993749i \(0.464390\pi\)
\(644\) 0 0
\(645\) −7.26622 −0.286107
\(646\) 0 0
\(647\) 39.7534 + 11.6726i 1.56287 + 0.458899i 0.944915 0.327317i \(-0.106145\pi\)
0.617952 + 0.786216i \(0.287963\pi\)
\(648\) 0 0
\(649\) 27.9855 17.9852i 1.09853 0.705979i
\(650\) 0 0
\(651\) 1.30783 9.09618i 0.0512580 0.356507i
\(652\) 0 0
\(653\) 2.41563 + 1.55243i 0.0945311 + 0.0607514i 0.587052 0.809549i \(-0.300288\pi\)
−0.492521 + 0.870300i \(0.663925\pi\)
\(654\) 0 0
\(655\) 1.71128 + 3.74719i 0.0668653 + 0.146415i
\(656\) 0 0
\(657\) −1.39908 9.73082i −0.0545833 0.379635i
\(658\) 0 0
\(659\) −24.3652 + 7.15427i −0.949134 + 0.278691i −0.719426 0.694569i \(-0.755595\pi\)
−0.229708 + 0.973260i \(0.573777\pi\)
\(660\) 0 0
\(661\) −8.91404 + 10.2874i −0.346716 + 0.400132i −0.902145 0.431432i \(-0.858008\pi\)
0.555429 + 0.831564i \(0.312554\pi\)
\(662\) 0 0
\(663\) 28.6777 + 33.0958i 1.11375 + 1.28533i
\(664\) 0 0
\(665\) −2.53094 + 5.54199i −0.0981457 + 0.214909i
\(666\) 0 0
\(667\) 3.07365 + 6.42008i 0.119012 + 0.248587i
\(668\) 0 0
\(669\) 0.245183 0.536876i 0.00947933 0.0207568i
\(670\) 0 0
\(671\) −9.57914 11.0549i −0.369799 0.426771i
\(672\) 0 0
\(673\) 20.9153 24.1376i 0.806227 0.930436i −0.192478 0.981301i \(-0.561652\pi\)
0.998706 + 0.0508652i \(0.0161979\pi\)
\(674\) 0 0
\(675\) −3.98076 + 1.16886i −0.153219 + 0.0449893i
\(676\) 0 0
\(677\) −0.0795608 0.553358i −0.00305777 0.0212673i 0.988235 0.152940i \(-0.0488742\pi\)
−0.991293 + 0.131673i \(0.957965\pi\)
\(678\) 0 0
\(679\) 4.54788 + 9.95846i 0.174532 + 0.382171i
\(680\) 0 0
\(681\) 15.5461 + 9.99085i 0.595727 + 0.382850i
\(682\) 0 0
\(683\) −0.697683 + 4.85249i −0.0266961 + 0.185675i −0.998806 0.0488497i \(-0.984444\pi\)
0.972110 + 0.234525i \(0.0753535\pi\)
\(684\) 0 0
\(685\) −15.2109 + 9.77547i −0.581180 + 0.373502i
\(686\) 0 0
\(687\) 0.315063 + 0.0925108i 0.0120204 + 0.00352951i
\(688\) 0 0
\(689\) −22.9323 −0.873650
\(690\) 0 0
\(691\) −3.82259 −0.145418 −0.0727091 0.997353i \(-0.523164\pi\)
−0.0727091 + 0.997353i \(0.523164\pi\)
\(692\) 0 0
\(693\) 3.03910 + 0.892362i 0.115446 + 0.0338980i
\(694\) 0 0
\(695\) −0.458408 + 0.294601i −0.0173884 + 0.0111748i
\(696\) 0 0
\(697\) −9.75906 + 67.8757i −0.369651 + 2.57098i
\(698\) 0 0
\(699\) 15.6085 + 10.0310i 0.590369 + 0.379407i
\(700\) 0 0
\(701\) −8.61440 18.8629i −0.325361 0.712442i 0.674300 0.738457i \(-0.264445\pi\)
−0.999662 + 0.0260154i \(0.991718\pi\)
\(702\) 0 0
\(703\) −6.36799 44.2904i −0.240173 1.67044i
\(704\) 0 0
\(705\) 8.94656 2.62695i 0.336947 0.0989366i
\(706\) 0 0
\(707\) −10.2035 + 11.7754i −0.383741 + 0.442861i
\(708\) 0 0
\(709\) −13.8730 16.0103i −0.521012 0.601280i 0.432872 0.901455i \(-0.357500\pi\)
−0.953884 + 0.300175i \(0.902955\pi\)
\(710\) 0 0
\(711\) −5.10253 + 11.1730i −0.191360 + 0.419019i
\(712\) 0 0
\(713\) 4.51404 36.0139i 0.169052 1.34873i
\(714\) 0 0
\(715\) 5.98051 13.0955i 0.223659 0.489744i
\(716\) 0 0
\(717\) −3.18001 3.66993i −0.118760 0.137056i
\(718\) 0 0
\(719\) −18.3299 + 21.1538i −0.683589 + 0.788904i −0.986438 0.164135i \(-0.947517\pi\)
0.302849 + 0.953039i \(0.402062\pi\)
\(720\) 0 0
\(721\) −7.58950 + 2.22848i −0.282648 + 0.0829928i
\(722\) 0 0
\(723\) 3.81885 + 26.5607i 0.142025 + 0.987803i
\(724\) 0 0
\(725\) −2.55797 5.60117i −0.0950005 0.208022i
\(726\) 0 0
\(727\) 10.1968 + 6.55311i 0.378180 + 0.243041i 0.715896 0.698206i \(-0.246018\pi\)
−0.337717 + 0.941248i \(0.609655\pi\)
\(728\) 0 0
\(729\) −0.142315 + 0.989821i −0.00527092 + 0.0366601i
\(730\) 0 0
\(731\) 48.5029 31.1709i 1.79394 1.15290i
\(732\) 0 0
\(733\) 32.0885 + 9.42203i 1.18522 + 0.348011i 0.814182 0.580609i \(-0.197186\pi\)
0.371033 + 0.928620i \(0.379004\pi\)
\(734\) 0 0
\(735\) −5.09790 −0.188039
\(736\) 0 0
\(737\) 26.3391 0.970213
\(738\) 0 0
\(739\) −18.0190 5.29085i −0.662839 0.194627i −0.0670274 0.997751i \(-0.521351\pi\)
−0.595812 + 0.803124i \(0.703170\pi\)
\(740\) 0 0
\(741\) −27.3687 + 17.5888i −1.00541 + 0.646140i
\(742\) 0 0
\(743\) 0.518529 3.60645i 0.0190230 0.132308i −0.978097 0.208151i \(-0.933256\pi\)
0.997120 + 0.0758428i \(0.0241647\pi\)
\(744\) 0 0
\(745\) 13.7553 + 8.84000i 0.503956 + 0.323873i
\(746\) 0 0
\(747\) 2.16853 + 4.74842i 0.0793424 + 0.173736i
\(748\) 0 0
\(749\) −0.0791714 0.550649i −0.00289286 0.0201203i
\(750\) 0 0
\(751\) 14.6666 4.30650i 0.535192 0.157146i −0.00295814 0.999996i \(-0.500942\pi\)
0.538150 + 0.842849i \(0.319123\pi\)
\(752\) 0 0
\(753\) −0.809594 + 0.934321i −0.0295032 + 0.0340486i
\(754\) 0 0
\(755\) 8.75341 + 10.1020i 0.318569 + 0.367648i
\(756\) 0 0
\(757\) 9.93935 21.7642i 0.361252 0.791032i −0.638518 0.769607i \(-0.720452\pi\)
0.999770 0.0214250i \(-0.00682032\pi\)
\(758\) 0 0
\(759\) 12.0651 + 3.30655i 0.437936 + 0.120020i
\(760\) 0 0
\(761\) −10.8888 + 23.8433i −0.394720 + 0.864317i 0.603058 + 0.797697i \(0.293949\pi\)
−0.997778 + 0.0666199i \(0.978779\pi\)
\(762\) 0 0
\(763\) 7.30393 + 8.42919i 0.264420 + 0.305157i
\(764\) 0 0
\(765\) −4.42290 + 5.10430i −0.159910 + 0.184546i
\(766\) 0 0
\(767\) 73.1987 21.4931i 2.64305 0.776070i
\(768\) 0 0
\(769\) −3.09981 21.5597i −0.111782 0.777461i −0.966185 0.257850i \(-0.916986\pi\)
0.854403 0.519611i \(-0.173923\pi\)
\(770\) 0 0
\(771\) −0.0539740 0.118186i −0.00194382 0.00425638i
\(772\) 0 0
\(773\) −3.38153 2.17318i −0.121625 0.0781638i 0.478414 0.878135i \(-0.341212\pi\)
−0.600039 + 0.799971i \(0.704848\pi\)
\(774\) 0 0
\(775\) −4.46854 + 31.0794i −0.160515 + 1.11640i
\(776\) 0 0
\(777\) −8.40452 + 5.40126i −0.301510 + 0.193769i
\(778\) 0 0
\(779\) −48.8800 14.3525i −1.75131 0.514231i
\(780\) 0 0
\(781\) 17.3378 0.620394
\(782\) 0 0
\(783\) −1.48419 −0.0530406
\(784\) 0 0
\(785\) 0.321564 + 0.0944196i 0.0114771 + 0.00336998i
\(786\) 0 0
\(787\) −20.4488 + 13.1417i −0.728921 + 0.468449i −0.851730 0.523981i \(-0.824446\pi\)
0.122809 + 0.992430i \(0.460810\pi\)
\(788\) 0 0
\(789\) −0.0268550 + 0.186781i −0.000956065 + 0.00664957i
\(790\) 0 0
\(791\) −0.871313 0.559959i −0.0309803 0.0199098i
\(792\) 0 0
\(793\) −13.9353 30.5140i −0.494857 1.08359i
\(794\) 0 0
\(795\) −0.503339 3.50080i −0.0178516 0.124161i
\(796\) 0 0
\(797\) 14.3220 4.20532i 0.507311 0.148960i −0.0180536 0.999837i \(-0.505747\pi\)
0.525365 + 0.850877i \(0.323929\pi\)
\(798\) 0 0
\(799\) −48.4502 + 55.9145i −1.71404 + 1.97811i
\(800\) 0 0
\(801\) −2.44199 2.81821i −0.0862835 0.0995765i
\(802\) 0 0
\(803\) 10.6529 23.3266i 0.375933 0.823179i
\(804\) 0 0
\(805\) 5.37175 0.0972853i 0.189329 0.00342886i
\(806\) 0 0
\(807\) −8.91436 + 19.5197i −0.313800 + 0.687127i
\(808\) 0 0
\(809\) 20.2201 + 23.3352i 0.710901 + 0.820423i 0.990182 0.139786i \(-0.0446416\pi\)
−0.279281 + 0.960209i \(0.590096\pi\)
\(810\) 0 0
\(811\) 5.84825 6.74925i 0.205360 0.236998i −0.643722 0.765260i \(-0.722611\pi\)
0.849082 + 0.528262i \(0.177156\pi\)
\(812\) 0 0
\(813\) 3.38578 0.994156i 0.118745 0.0348666i
\(814\) 0 0
\(815\) −1.51009 10.5029i −0.0528961 0.367901i
\(816\) 0 0
\(817\) 17.7932 + 38.9617i 0.622506 + 1.36310i
\(818\) 0 0
\(819\) 6.11064 + 3.92707i 0.213523 + 0.137223i
\(820\) 0 0
\(821\) 2.47290 17.1994i 0.0863047 0.600263i −0.900070 0.435746i \(-0.856485\pi\)
0.986374 0.164517i \(-0.0526064\pi\)
\(822\) 0 0
\(823\) 17.6060 11.3147i 0.613707 0.394405i −0.196539 0.980496i \(-0.562970\pi\)
0.810245 + 0.586091i \(0.199334\pi\)
\(824\) 0 0
\(825\) −10.3839 3.04898i −0.361520 0.106152i
\(826\) 0 0
\(827\) −17.6637 −0.614226 −0.307113 0.951673i \(-0.599363\pi\)
−0.307113 + 0.951673i \(0.599363\pi\)
\(828\) 0 0
\(829\) 32.0581 1.11343 0.556713 0.830705i \(-0.312062\pi\)
0.556713 + 0.830705i \(0.312062\pi\)
\(830\) 0 0
\(831\) 17.8951 + 5.25448i 0.620775 + 0.182276i
\(832\) 0 0
\(833\) 34.0291 21.8691i 1.17904 0.757721i
\(834\) 0 0
\(835\) 0.618603 4.30248i 0.0214076 0.148893i
\(836\) 0 0
\(837\) 6.36676 + 4.09167i 0.220067 + 0.141429i
\(838\) 0 0
\(839\) 9.20564 + 20.1575i 0.317814 + 0.695916i 0.999357 0.0358627i \(-0.0114179\pi\)
−0.681543 + 0.731778i \(0.738691\pi\)
\(840\) 0 0
\(841\) 3.81364 + 26.5244i 0.131505 + 0.914635i
\(842\) 0 0
\(843\) 26.0896 7.66061i 0.898575 0.263845i
\(844\) 0 0
\(845\) 13.7660 15.8868i 0.473566 0.546524i
\(846\) 0 0
\(847\) −3.33624 3.85023i −0.114635 0.132295i
\(848\) 0 0
\(849\) 1.54319 3.37912i 0.0529623 0.115971i
\(850\) 0 0
\(851\) −32.8028 + 21.9304i −1.12447 + 0.751764i
\(852\) 0 0
\(853\) −19.5118 + 42.7250i −0.668073 + 1.46288i 0.206731 + 0.978398i \(0.433718\pi\)
−0.874804 + 0.484478i \(0.839010\pi\)
\(854\) 0 0
\(855\) −3.28578 3.79199i −0.112371 0.129683i
\(856\) 0 0
\(857\) 6.20842 7.16489i 0.212075 0.244748i −0.639738 0.768593i \(-0.720957\pi\)
0.851814 + 0.523845i \(0.175503\pi\)
\(858\) 0 0
\(859\) −30.2143 + 8.87171i −1.03090 + 0.302699i −0.753076 0.657934i \(-0.771431\pi\)
−0.277822 + 0.960633i \(0.589613\pi\)
\(860\) 0 0
\(861\) 1.61873 + 11.2585i 0.0551660 + 0.383688i
\(862\) 0 0
\(863\) −14.4440 31.6278i −0.491678 1.07662i −0.979085 0.203450i \(-0.934785\pi\)
0.487407 0.873175i \(-0.337943\pi\)
\(864\) 0 0
\(865\) 6.03077 + 3.87574i 0.205052 + 0.131779i
\(866\) 0 0
\(867\) 5.20741 36.2183i 0.176853 1.23004i
\(868\) 0 0
\(869\) −26.9540 + 17.3223i −0.914353 + 0.587619i
\(870\) 0 0
\(871\) 57.9561 + 17.0174i 1.96377 + 0.576614i
\(872\) 0 0
\(873\) −9.01605 −0.305147
\(874\) 0 0
\(875\) −10.2492 −0.346485
\(876\) 0 0
\(877\) 22.3108 + 6.55106i 0.753384 + 0.221213i 0.635805 0.771850i \(-0.280668\pi\)
0.117579 + 0.993064i \(0.462487\pi\)
\(878\) 0 0
\(879\) 14.2379 9.15017i 0.480234 0.308628i
\(880\) 0 0
\(881\) −7.44501 + 51.7812i −0.250829 + 1.74455i 0.342433 + 0.939542i \(0.388749\pi\)
−0.593262 + 0.805010i \(0.702160\pi\)
\(882\) 0 0
\(883\) 1.81544 + 1.16672i 0.0610946 + 0.0392631i 0.570832 0.821067i \(-0.306621\pi\)
−0.509737 + 0.860330i \(0.670257\pi\)
\(884\) 0 0
\(885\) 4.88773 + 10.7026i 0.164299 + 0.359765i
\(886\) 0 0
\(887\) 1.82373 + 12.6844i 0.0612350 + 0.425899i 0.997261 + 0.0739672i \(0.0235660\pi\)
−0.936026 + 0.351932i \(0.885525\pi\)
\(888\) 0 0
\(889\) −15.5849 + 4.57613i −0.522700 + 0.153479i
\(890\) 0 0
\(891\) −1.70822 + 1.97139i −0.0572274 + 0.0660439i
\(892\) 0 0
\(893\) −35.9938 41.5390i −1.20449 1.39005i
\(894\) 0 0
\(895\) 5.66780 12.4108i 0.189454 0.414846i
\(896\) 0 0
\(897\) 24.4115 + 15.0708i 0.815077 + 0.503200i
\(898\) 0 0
\(899\) −4.66620 + 10.2175i −0.155626 + 0.340774i
\(900\) 0 0
\(901\) 18.3777 + 21.2090i 0.612250 + 0.706574i
\(902\) 0 0
\(903\) 6.26260 7.22743i 0.208406 0.240514i
\(904\) 0 0
\(905\) −16.7351 + 4.91386i −0.556292 + 0.163342i
\(906\) 0 0
\(907\) −1.27805 8.88901i −0.0424368 0.295155i −0.999977 0.00685318i \(-0.997819\pi\)
0.957540 0.288302i \(-0.0930905\pi\)
\(908\) 0 0
\(909\) −5.33054 11.6723i −0.176803 0.387144i
\(910\) 0 0
\(911\) 15.0425 + 9.66724i 0.498381 + 0.320290i 0.765568 0.643355i \(-0.222458\pi\)
−0.267187 + 0.963645i \(0.586094\pi\)
\(912\) 0 0
\(913\) −1.93788 + 13.4783i −0.0641345 + 0.446066i
\(914\) 0 0
\(915\) 4.35235 2.79709i 0.143884 0.0924688i
\(916\) 0 0
\(917\) −5.20210 1.52747i −0.171788 0.0504416i
\(918\) 0 0
\(919\) 52.1924 1.72167 0.860835 0.508885i \(-0.169942\pi\)
0.860835 + 0.508885i \(0.169942\pi\)
\(920\) 0 0
\(921\) 28.7453 0.947189
\(922\) 0 0
\(923\) 38.1497 + 11.2018i 1.25571 + 0.368711i
\(924\) 0 0
\(925\) 28.7161 18.4547i 0.944181 0.606788i
\(926\) 0 0
\(927\) 0.927067 6.44789i 0.0304489 0.211777i
\(928\) 0 0
\(929\) −22.6910 14.5826i −0.744467 0.478440i 0.112603 0.993640i \(-0.464081\pi\)
−0.857070 + 0.515200i \(0.827718\pi\)
\(930\) 0 0
\(931\) 12.4835 + 27.3351i 0.409131 + 0.895871i
\(932\) 0 0
\(933\) 3.67621 + 25.5686i 0.120354 + 0.837078i
\(934\) 0 0
\(935\) −16.9041 + 4.96351i −0.552825 + 0.162324i
\(936\) 0 0
\(937\) 25.9973 30.0025i 0.849296 0.980140i −0.150668 0.988584i \(-0.548143\pi\)
0.999964 + 0.00844436i \(0.00268796\pi\)
\(938\) 0 0
\(939\) 7.63294 + 8.80888i 0.249092 + 0.287467i
\(940\) 0 0
\(941\) −16.0338 + 35.1090i −0.522686 + 1.14452i 0.445727 + 0.895169i \(0.352945\pi\)
−0.968413 + 0.249353i \(0.919782\pi\)
\(942\) 0 0
\(943\) 7.19746 + 44.3435i 0.234382 + 1.44402i
\(944\) 0 0
\(945\) −0.465378 + 1.01903i −0.0151387 + 0.0331492i
\(946\) 0 0
\(947\) −23.9757 27.6694i −0.779106 0.899136i 0.217939 0.975962i \(-0.430067\pi\)
−0.997044 + 0.0768267i \(0.975521\pi\)
\(948\) 0 0
\(949\) 38.5116 44.4447i 1.25014 1.44274i
\(950\) 0 0
\(951\) −14.8939 + 4.37325i −0.482968 + 0.141812i
\(952\) 0 0
\(953\) −6.45166 44.8723i −0.208990 1.45356i −0.776462 0.630164i \(-0.782988\pi\)
0.567472 0.823393i \(-0.307921\pi\)
\(954\) 0 0
\(955\) 0.141312 + 0.309430i 0.00457275 + 0.0100129i
\(956\) 0 0
\(957\) −3.25694 2.09311i −0.105282 0.0676606i
\(958\) 0 0
\(959\) 3.38670 23.5550i 0.109362 0.760631i
\(960\) 0 0
\(961\) 22.1060 14.2066i 0.713096 0.458279i
\(962\) 0 0
\(963\) 0.439592 + 0.129076i 0.0141656 + 0.00415941i
\(964\) 0 0
\(965\) 18.3376 0.590308
\(966\) 0 0
\(967\) 8.82892 0.283919 0.141959 0.989872i \(-0.454660\pi\)
0.141959 + 0.989872i \(0.454660\pi\)
\(968\) 0 0
\(969\) 38.2000 + 11.2165i 1.22716 + 0.360327i
\(970\) 0 0
\(971\) −6.37284 + 4.09558i −0.204514 + 0.131433i −0.638889 0.769299i \(-0.720606\pi\)
0.434375 + 0.900732i \(0.356969\pi\)
\(972\) 0 0
\(973\) 0.102064 0.709871i 0.00327202 0.0227574i
\(974\) 0 0
\(975\) −20.8786 13.4178i −0.668649 0.429715i
\(976\) 0 0
\(977\) −8.42213 18.4419i −0.269448 0.590008i 0.725743 0.687966i \(-0.241496\pi\)
−0.995191 + 0.0979579i \(0.968769\pi\)
\(978\) 0 0
\(979\) −1.38433 9.62821i −0.0442433 0.307719i
\(980\) 0 0
\(981\) −8.81331 + 2.58782i −0.281387 + 0.0826228i
\(982\) 0 0
\(983\) −13.7231 + 15.8373i −0.437698 + 0.505131i −0.931147 0.364644i \(-0.881191\pi\)
0.493449 + 0.869775i \(0.335736\pi\)
\(984\) 0 0
\(985\) 2.23928 + 2.58427i 0.0713495 + 0.0823417i
\(986\) 0 0
\(987\) −5.09793 + 11.1629i −0.162269 + 0.355319i
\(988\) 0 0
\(989\) 24.2139 28.9887i 0.769956 0.921786i
\(990\) 0 0
\(991\) −11.5706 + 25.3360i −0.367551 + 0.804825i 0.632003 + 0.774966i \(0.282233\pi\)
−0.999554 + 0.0298588i \(0.990494\pi\)
\(992\) 0 0
\(993\) −20.6121 23.7876i −0.654104 0.754876i
\(994\) 0 0
\(995\) 4.39454 5.07156i 0.139316 0.160779i
\(996\) 0 0
\(997\) 37.6349 11.0506i 1.19191 0.349976i 0.375154 0.926962i \(-0.377590\pi\)
0.816754 + 0.576987i \(0.195772\pi\)
\(998\) 0 0
\(999\) −1.17092 8.14390i −0.0370461 0.257662i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 552.2.q.d.121.1 yes 30
23.4 even 11 inner 552.2.q.d.73.1 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
552.2.q.d.73.1 30 23.4 even 11 inner
552.2.q.d.121.1 yes 30 1.1 even 1 trivial